Profiling & Parallelization

Lecture 21

Dr. Colin Rundel

Profiling & Benchmarking

profvis demo

n = 1e6
d = tibble(
  x1 = rt(n, df = 3),
  x2 = rt(n, df = 3),
  x3 = rt(n, df = 3),
  x4 = rt(n, df = 3),
  x5 = rt(n, df = 3),
) |>
  mutate(y = -2*x1 - 1*x2 + 0*x3 + 1*x4 + 2*x5 + rnorm(n))
profvis::profvis(lm(y~., data=d))

profvis demo 2

profvis::profvis({
  data = data.frame(value = runif(5e4))

  data$sum[1] = data$value[1]
  for (i in seq(2, nrow(data))) {
    data$sum[i] = data$sum[i-1] + data$value[i]
  }
})
profvis::profvis({
  x = runif(5e4)
  sum = x[1]
  for (i in seq(2, length(x))) {
    sum[i] = sum[i-1] + x[i]
  }
})

Benchmarking - bench

d = tibble(
  x = runif(10000),
  y = runif(10000)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))
# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]           128µs    138µs     7179.  238.91KB     19.4
2 d[which(d$x > 0.5), ]    137µs    152µs     6514.  269.81KB     33.9
3 subset(d, x > 0.5)       168µs    190µs     5138.  287.91KB     26.3
4 filter(d, x > 0.5)       385µs    412µs     2384.    1.47MB     42.3

Larger n

d = tibble(
  x = runif(1e6),
  y = runif(1e6)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))
# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]          11.6ms   12.2ms      82.5    13.4MB     78.2
2 d[which(d$x > 0.5), ]   12.8ms   13.6ms      74.0    24.8MB    123. 
3 subset(d, x > 0.5)      17.3ms   17.8ms      56.1    24.8MB     84.2
4 filter(d, x > 0.5)        14ms   14.2ms      70.4    24.8MB     51.9

bench - relative results

summary(b, relative=TRUE)
# A tibble: 4 × 6
  expression              min median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <dbl>  <dbl>     <dbl>     <dbl>    <dbl>
1 d[d$x > 0.5, ]         1      1         1.47      1        1.51
2 d[which(d$x > 0.5), ]  1.10   1.12      1.32      1.86     2.38
3 subset(d, x > 0.5)     1.49   1.46      1         1.86     1.62
4 filter(d, x > 0.5)     1.20   1.17      1.25      1.86     1   

t.test

Imagine we have run 1000 experiments (rows), each of which collects data on 50 individuals (columns). The first 25 individuals in each experiment are assigned to group 1 and the rest to group 2.

The goal is to calculate the t-statistic for each experiment comparing group 1 to group 2.

m = 1000
n = 50
X = matrix(
  rnorm(m * n, mean = 10, sd = 3), 
  ncol = m
) |>
  as.data.frame() |>
  set_names(paste0("exp", seq_len(m))) |>
  mutate(
    ind = seq_len(n),
    group = rep(1:2, each = n/2)
  ) |>
  as_tibble() |>
  relocate(ind, group)
X
# A tibble: 50 × 1,002
     ind group  exp1  exp2  exp3  exp4  exp5  exp6
   <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
 1     1     1  9.54 10.9  11.8  12.2  12.2  12.6 
 2     2     1  7.78 12.7   9.21  9.48 13.1   7.67
 3     3     1 10.5  10.8   7.36  7.04 12.1   6.23
 4     4     1  8.71  7.36  9.20 12.4   4.42 12.8 
 5     5     1  7.38  9.47 10.5  13.4  10.2   7.35
 6     6     1  9.68  9.71 10.9   8.89 10.9  13.3 
 7     7     1  7.40 13.2   6.02  8.76  7.02  3.30
 8     8     1 12.7  14.2   3.93  8.53  5.45  6.96
 9     9     1  7.15  8.81 13.8   8.28  8.64  7.75
10    10     1  8.80 14.4  15.9   7.93 12.3  13.6 
# ℹ 40 more rows
# ℹ 994 more variables: exp7 <dbl>, exp8 <dbl>,
#   exp9 <dbl>, exp10 <dbl>, exp11 <dbl>,
#   exp12 <dbl>, exp13 <dbl>, exp14 <dbl>,
#   exp15 <dbl>, exp16 <dbl>, exp17 <dbl>,
#   exp18 <dbl>, exp19 <dbl>, exp20 <dbl>,
#   exp21 <dbl>, exp22 <dbl>, exp23 <dbl>, …

Implementations

ttest_formula = function(X, m) {
  for(i in 1:m) t.test(X[[2+i]] ~ X$group)$stat
}
system.time(ttest_formula(X,m))
   user  system elapsed 
  0.186   0.001   0.187 
ttest_for = function(X, m) {
  for(i in 1:m) t.test(X[[2+i]][X$group == 1], X[[2+i]][X$group == 2])$stat
}
system.time(ttest_for(X,m))
   user  system elapsed 
  0.066   0.001   0.068 
ttest_apply = function(X) {
  f = function(x, g) {
    t.test(x[g==1], x[g==2])$stat
  }
  apply(X[,-(1:2)], 2, f, X$group)
}
system.time(ttest_apply(X))
   user  system elapsed 
  0.051   0.001   0.052 

Implementations (cont.)

ttest_hand_calc = function(X) {
  f = function(x, grp) {
    t_stat = function(x) {
      m = mean(x)
      n = length(x)
      var = sum((x - m) ^ 2) / (n - 1)
      
      list(m = m, n = n, var = var)
    }
    
    g1 = t_stat(x[grp == 1])
    g2 = t_stat(x[grp == 2])
    
    se_total = sqrt(g1$var / g1$n + g2$var / g2$n)
    (g1$m - g2$m) / se_total
  }
  
    apply(X[,-(1:2)], 2, f, X$group)
}
system.time(ttest_hand_calc(X))
   user  system elapsed 
  0.015   0.000   0.015 

Comparison

bench::mark(
  ttest_formula(X, m),
  ttest_for(X, m),
  ttest_apply(X),
  ttest_hand_calc(X),
  check=FALSE
)
Warning: Some expressions had a GC in every iteration; so filtering
is disabled.
# A tibble: 4 × 6
  expression               min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>          <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 ttest_formula(X, m) 188.85ms 192.35ms      5.18    8.24MB     22.5
2 ttest_for(X, m)      61.45ms  64.52ms     15.6     1.91MB     23.5
3 ttest_apply(X)       54.61ms  57.83ms     17.3     3.48MB     23.0
4 ttest_hand_calc(X)    8.73ms   9.04ms     93.1     3.44MB     23.8

Parallelization

parallel

Part of the base packages in R

  • tools for the forking of R processes (some functions do not work on Windows)

  • Core functions:

    • detectCores

    • pvec

    • mclapply

    • mcparallel & mccollect

detectCores

Surprisingly, detects the number of cores of the current system.

detectCores()
[1] 10

pvec

Parallelization of a vectorized function call

system.time(pvec(1:1e7, sqrt, mc.cores = 1))
   user  system elapsed 
  0.017   0.012   0.028 
system.time(pvec(1:1e7, sqrt, mc.cores = 4))
   user  system elapsed 
  0.176   0.174   0.278 
system.time(pvec(1:1e7, sqrt, mc.cores = 8))
   user  system elapsed 
  0.090   0.201   0.178 
system.time(sqrt(1:1e7))
   user  system elapsed 
  0.017   0.020   0.037 

pvec - bench::system_time

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 1))
process    real 
 63.2ms  62.3ms 
bench::system_time(pvec(1:1e7, sqrt, mc.cores = 4))
process    real 
  163ms   188ms 
bench::system_time(pvec(1:1e7, sqrt, mc.cores = 8))
process    real 
  219ms   230ms 

bench::system_time(Sys.sleep(.5))
process    real 
  142µs   493ms 
system.time(Sys.sleep(.5))
   user  system elapsed 
  0.000   0.000   0.507 

Cores by size

cores = c(1,4,6,8,10)
order = 6:8
f = function(x,y) {
  system.time(
    pvec(1:(10^y), sqrt, mc.cores = x)
  )[3]
}

res = map(
  cores, 
  function(x) {
     map_dbl(order, f, x = x)
  }
) |> 
  do.call(rbind, args = _)

rownames(res) = paste0(cores," cores")
colnames(res) = paste0("10^",order)
res
          10^6  10^7  10^8
1 cores  0.004 0.036 0.367
4 cores  0.036 0.151 1.917
6 cores  0.030 0.111 1.332
8 cores  0.042 0.155 1.506
10 cores 0.045 0.192 1.567

mclapply

Parallelized version of lapply

system.time(rnorm(1e7))
   user  system elapsed 
  0.264   0.003   0.267 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 2)))
   user  system elapsed 
  0.329   0.112   0.284 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 4)))
   user  system elapsed 
  0.334   0.102   0.180 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 8)))
   user  system elapsed 
  0.340   0.150   0.174 
system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 10)))
   user  system elapsed 
  0.362   0.162   0.178 

mcparallel

Asynchronously evaluation of an R expression in a separate process

m = mcparallel(rnorm(1e6))
n = mcparallel(rbeta(1e6,1,1))
o = mcparallel(rgamma(1e6,1,1))
str(m)
List of 2
 $ pid: int 78380
 $ fd : int [1:2] 5 8
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"
str(n)
List of 2
 $ pid: int 78381
 $ fd : int [1:2] 6 10
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"

mccollect

Checks mcparallel objects for completion

str(mccollect(list(m,n,o)))
List of 3
 $ 78380: num [1:1000000] -0.1712 -0.0412 -0.9246 -1.2699 -1.274 ...
 $ 78381: num [1:1000000] 0.201 0.838 0.289 0.192 0.722 ...
 $ 78382: num [1:1000000] 0.7425 0.5719 0.4059 0.0538 0.7154 ...

mccollect - waiting

p = mcparallel(mean(rnorm(1e5)))
mccollect(p, wait = FALSE, 10)
$`78383`
[1] 0.006051739
mccollect(p, wait = FALSE)
Warning in selectChildren(jobs, timeout): cannot wait for child 78383
as it does not exist
NULL
mccollect(p, wait = FALSE)
Warning in selectChildren(jobs, timeout): cannot wait for child 78383
as it does not exist
NULL

doMC & foreach

doMC & foreach

Packages by Revolution Analytics that provides the foreach function which is a parallelizable for loop (and then some).

  • Core functions:

    • registerDoMC

    • foreach, %dopar%, %do%

registerDoMC

Primarily used to set the number of cores used by foreach, by default uses options("cores") or half the number of cores found by detectCores from the parallel package.

options("cores")
$cores
NULL
detectCores()
[1] 10
getDoParWorkers()
[1] 1
registerDoMC(4)
getDoParWorkers()
[1] 4

foreach

A slightly more powerful version of base for loops (think for with an lapply flavor). Combined with %do% or %dopar% for single or multicore execution.

for(i in 1:10) {
  sqrt(i)
}
foreach(i = 1:5) %do% {
  sqrt(i)   
}
[[1]]
[1] 1

[[2]]
[1] 1.414214

[[3]]
[1] 1.732051

[[4]]
[1] 2

[[5]]
[1] 2.236068

foreach - iterators

foreach can iterate across more than one value, but it doesn’t do length coercion

foreach(i = 1:5, j = 1:5) %do% {
  sqrt(i^2+j^2)   
}
[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

[[3]]
[1] 4.242641

[[4]]
[1] 5.656854

[[5]]
[1] 7.071068
foreach(i = 1:5, j = 1:2) %do% {
  sqrt(i^2+j^2)   
}
[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

foreach - combining results

foreach(i = 1:5, .combine='c') %do% {
  sqrt(i)
}
[1] 1.000000 1.414214 1.732051 2.000000 2.236068
foreach(i = 1:5, .combine='cbind') %do% {
  sqrt(i)
}
     result.1 result.2 result.3 result.4 result.5
[1,]        1 1.414214 1.732051        2 2.236068
foreach(i = 1:5, .combine='+') %do% {
  sqrt(i)
}
[1] 8.382332

foreach - parallelization

Swapping out %do% for %dopar% will use the parallel backend.

registerDoMC(4)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))
   user  system elapsed 
  0.297   0.040   0.113 
registerDoMC(8)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))
   user  system elapsed 
  0.303   0.052   0.082 
registerDoMC(10)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))
   user  system elapsed 
  0.327   0.064   0.078 

furrr / future

system.time( purrr::map(c(1,1,1), Sys.sleep) )
   user  system elapsed 
  0.001   0.001   3.012 
system.time( furrr::future_map(c(1,1,1), Sys.sleep) )
   user  system elapsed 
  0.042   0.006   3.076 
future::plan(future::multisession) # See also future::multicore
system.time( furrr::future_map(c(1,1,1), Sys.sleep) )
   user  system elapsed 
  0.202   0.006   1.459 

Example - Bootstraping

Bootstrapping is a resampling scheme where the original data is repeatedly reconstructed by taking a samples of size n (with replacement) from the original data, and using that to repeat an analysis procedure of interest. Below is an example of fitting a local regression (loess) to some synthetic data, we will construct a bootstrap prediction interval for this model.

set.seed(3212016)
d = data.frame(x = 1:120) |>
    mutate(y = sin(2*pi*x/120) + runif(length(x),-1,1))

l = loess(y ~ x, data=d)
p = predict(l, se=TRUE)

d = d |> mutate(
  pred_y = p$fit,
  pred_y_se = p$se.fit
)

ggplot(d, aes(x,y)) +
  geom_point(color="gray50") +
  geom_ribbon(
    aes(ymin = pred_y - 1.96 * pred_y_se, 
        ymax = pred_y + 1.96 * pred_y_se), 
    fill="red", alpha=0.25
  ) +
  geom_line(aes(y=pred_y)) +
  theme_bw()

Bootstraping Demo

What to use when?

Optimal use of parallelization / multiple cores is hard, there isn’t one best solution

  • Don’t underestimate the overhead cost

  • Experimentation is key

  • Measure it or it didn’t happen

  • Be aware of the trade off between developer time and run time

BLAS and LAPACK

Statistics and Linear Algebra

An awful lot of statistics is at its core linear algebra.

For example:

  • Linear regession models, find

\[ \hat{\beta} = (X^T X)^{-1} X^Ty \]

  • Principle component analysis

    • Find \(T = XW\) where \(W\) is a matrix whose columns are the eigenvectors of \(X^TX\).

    • Often solved via SVD - Let \(X = U\Sigma W^T\) then \(T = U\Sigma\).

Numerical Linear Algebra

Not unique to Statistics, these are the type of problems that come up across all areas of numerical computing.

  • Numerical linear algebra \(\ne\) mathematical linear algebra

  • Efficiency and stability of numerical algorithms matter

    • Designing and implementing these algorithms is hard
  • Don’t reinvent the wheel - common core linear algebra tools (well defined API)

BLAS and LAPACK

Low level algorithms for common linear algebra operations

BLAS

  • Basic Linear Algebra Subprograms

  • Copying, scaling, multiplying vectors and matrices

  • Origins go back to 1979, written in Fortran

LAPACK

  • Linear Algebra Package

  • Higher level functionality building on BLAS.

  • Linear solvers, eigenvalues, and matrix decompositions

  • Origins go back to 1992, mostly Fortran (expanded on LINPACK, EISPACK)

Modern variants?

Most default BLAS and LAPACK implementations (like R’s defaults) are somewhat dated

  • Written in Fortran and designed for a single cpu core

  • Certain (potentially non-optimal) hard coded defaults (e.g. block size).

Multithreaded alternatives:

  • ATLAS - Automatically Tuned Linear Algebra Software

  • OpenBLAS - fork of GotoBLAS from TACC at UTexas

  • Intel MKL - Math Kernel Library, part of Intel’s commercial compiler tools

  • cuBLAS / Magma - GPU libraries from Nvidia and UTK respectively

  • Accelerate / vecLib - Apple’s framework for GPU and multicore computing

OpenBLAS Matrix Multiply Performance

x=matrix(runif(5000^2),ncol=5000)

sizes = c(100,500,1000,2000,3000,4000,5000)
cores = c(1,2,4,8,16)

sapply(
  cores, 
  function(n_cores) {
    flexiblas::flexiblas_set_num_threads(n_cores)
    sapply(
      sizes, 
      function(s) {
        y = x[1:s,1:s]
        system.time(y %*% y)[3]
      }
    )
  }
)

n 1 core 2 cores 4 cores 8 cores 16 cores
100 0.000 0.000 0.000 0.000 0.000
500 0.004 0.003 0.002 0.002 0.004
1000 0.028 0.016 0.010 0.007 0.009
2000 0.207 0.110 0.058 0.035 0.039
3000 0.679 0.352 0.183 0.103 0.081
4000 1.587 0.816 0.418 0.227 0.145
5000 3.104 1.583 0.807 0.453 0.266